Chapter IV: «the Theory of Ionization; Ionization and Electrical Conductivity» (1)

Of the laws and hypotheses concerning gases, the one that is perhaps of most importance to chemistry is Avogadro's hypothesis. With the aid of this hypothesis, we are able to determine the relative molecular weights[49] of such elements and compounds as are gases, or are volatile at higher temperatures. If equal volumes of gases, under the same conditions of temperature and pressure, contain the same number of molecules, then the weights of such equal volumes also represent the relative weights of the molecules composing the gases. As a standard, for expressing the relative molecular weights in definite numbers, the molecular weight of oxygen is taken by convention to be 32, and all other molecular weights are expressed in terms of this standard. The density, or weight of one liter of oxygen at 0° and 760 mm., is 1.429 grams, and the ‹molecular weight expressed in grams (molar weight)› of oxygen, 32 grams, occupies, therefore, 32 / 1.429, or 22.4 liters. The weights of ‹this same volume›, 22.4 liters, of gases and vapors, calculated for 0° and 760 mm. pressure,[50] express then directly, in terms of the oxygen standard, the relative molecular weights of the elements or compounds forming the gases. The weights themselves give us directly their gram-molecular or molar weights.

When molecular weights are determined in this way, with the aid of Avogadro's hypothesis, results are obtained which agree [p034] perfectly with the chemical behavior of the compounds or elements in question. The molecular weights of hydrogen chloride, water, ammonia, and marsh gas, for instance, are found to be 36.5, 18, 17 and 16, respectively, corresponding to the formulæ[51] HCl, H_{2}O, NH_{3} and CH_{4}, and in confirmation of these results we find, by methods used especially in organic chemistry, that these compounds show a chemical behavior agreeing perfectly with the presence of one, two, three and four hydrogen atoms, respectively, in their molecules. Marsh gas, for instance, by treatment with chlorine, yields a monochloride, CH_{3}Cl, a dichloride, CH_{2}Cl_{2}, a trichloride (chloroform), CHCl_{3}, and a tetrachloride, CCl_{4}. Water, by proper treatment, may be converted in successive stages into alcohol, (C_{2}H_{5})OH, and then into ether, (C_{2}H_{5})O(C_{2}H_{5}), or into sodium hydroxide, NaOH, and sodium oxide, Na_{2}O.

It is this perfect agreement between the chemical behavior and the formulæ (as based on these molecular weights and on the analysis of compounds), which forms the strongest ‹experimental evidence› of the correctness of the fundamental assumption of Avogadro's hypothesis. The agreement has been shown to hold for innumerable compounds, even for those of greatest complexity, and it was such agreement which finally led to the general acceptance of the hypothesis. The experimental evidence of this nature is so strong, so extensive and so completely corroborative of the hypothesis, that many chemists, rather justly, consider the hypothesis to have been established as a law, although the evidence is circumstantial rather than direct.

While the application of Avogadro's hypothesis thus gives results agreeing well with the observed chemical behavior of very many important compounds, observations have been made which, at first sight, do not appear to agree with the requirements of the hypothesis and which seem to raise a doubt as to the ‹universal truth› of its fundamental assumption. Thus, if equal volumes of hydrogen chloride and ammonia, of the same temperature and pressure, are brought together, ammonium chloride is formed, both gases being totally consumed. Since, according to the hypothesis, equal volumes, under the conditions obtaining, contain the same numbers of molecules, the formation of ammonium chloride takes place according to the equation NH_{3} + HCl → NH_{4}Cl, and we should anticipate that the molecular weight of [p035] ammonium chloride would be 17 + 36.5 or 53.5. However, when the molecular weight is determined by obtaining the weight of a measured volume of ammonium chloride vapor, at a temperature sufficiently high to vaporize the salt, and the observations are reduced to standard conditions of temperature and pressure, 26.75 grams is found as the calculated weight of 22.4 liters, and this weight, according to this hypothesis, should be the molecular weight of the chloride. This contradiction in two conclusions, each reached by the application of Avogadro's hypothesis to experimental observations, would, at the first glance, make one hesitate to accept the hypothesis as representing a universal truth; it might seem as if in some gases, such as ammonium chloride vapor, there might be only half as many molecules in a given volume as in the same volume of the majority of gases.

«Gaseous Dissociation.»—The discrepancy between the two conclusions and any doubt as to the universal soundness of the great generalization expressed in Avogadro's hypothesis disappear, however, in the light of a closer study of the composition of ammonium chloride vapor. It was suggested simultaneously by Cannizzaro, by Kopp and by Kékulé[52] that the abnormally low result, obtained for the molecular weight of ammonium chloride from a study of its vapor density, is due to the ‹dissociation› of the salt at high temperatures into its components, ammonia and hydrogen chloride, the ‹average› of whose ‹molecular weights› is, in fact, (17 + 36.5) / 2, or 26.75, the value found experimentally for the vapor of ammonium chloride. Proof of the correctness of this interpretation was furnished by Pébal,[53] who showed that ammonium chloride vapor does consist of the two gases, the lighter of which, ammonia, diffuses more rapidly through porous walls (Pébal used an asbestos stopper) than does the heavier, hydrogen chloride. The dissociation may be easily demonstrated by using an air cushion as a porous wall.[54] From the mixture produced by vaporizing ammonium chloride,[55] the ammonia will diffuse more rapidly through the layer of air than will the hydrogen [p036] chloride, and the gases may be recognized in succession by their action on litmus paper (‹exp.›).

The ‹gaseous dissociation› of other ammonium salts, of phosphorus pentachloride and pentabromide (PX_{5} ⇄ PX_{3} + X_{2}), and of a number of less common compounds, has been demonstrated in similar ways. As a result of the study of each case, the important conclusion has been reached that, as far as our knowledge goes, there are no exceptions to Avogadro's hypothesis, and this hypothesis seems therefore to represent a universal truth.[56]

«Molecular Weight Determinations in Solution.»—Van 't Hoff's extension of the Avogadro Hypothesis, so that it shall apply to solutes in dilute solutions, is the basis of another general method of greatest value for determining molecular weights. Equal volumes of dilute solutions of the same osmotic pressure and the same temperature contain, according to van 't Hoff, the same numbers of dissolved molecules, irrespective of the solvent used. Furthermore, the number of dissolved molecules is identical with that which a gas of the same pressure and at the same temperature would contain in the same volume. To determine the molecular weight of a solute, therefore, we may calculate, from the osmotic pressure, the temperature and the concentration of the solution,[57] that weight of the solute which, in 22.4 liters of the solution, at 0° would give 760 mm. osmotic pressure; the weight found represents, in grams, the ‹molecular weight› of the solute ‹in the solution used›. [p037]

The fact that ‹all solvents› and ‹all solutes› are included in this hypothesis, with the sole limiting condition that the solution must be dilute, is one of great significance and of greatest practical importance, as we may use any suitable solvent for determinations.

When molecular weights are determined in this way, a very large number of compounds give the same molecular weight by the solution method as by the gas method. For instance we have:

Substance. Mol. Wt. Mol. Wt. Solvent.
Gas Method. Sol. Method.
Chloroform, CHCl_{3} 119.5 119.5 Benzene
Carbon bisulphide, CS_{2} 76 76 Benzene
Methyl (wood) alcohol, CH_{4}O 32 32 Water
Ethyl (ordinary) alcohol,
C_{2}H_{6}O 46 46 Water
Ether, C_{4}H_{10}O 74 74 Acet. Acid

Further, the molecular weight of glucose is found in aqueous solutions to be 180, conforming to the formula C_{6}H_{12}O_{6}, and agreeing with the molecular weight as obtained by a chemical study of compounds derived from glucose.

While there are, then, very many agreements in the molecular weights determined by the solution and by the older methods, it was recognized, at the outset,[58] that there is also a large number of apparently ‹abnormal› cases, in which, in particular, ‹much lower molecular weights› are obtained by the solution methods than by the gas method,—lower even than the weights consistent with the accepted atomic weights of the elements in the compounds in question.[59] For instance, we find 36.5 to be the molecular weight of hydrogen chloride in the gas form, but in ‹aqueous› solution its apparent molecular weight, as determined on the basis of van 't Hoff's hypothesis, is not even a constant; it is found to be less than 36.5 and approaches the limit 18.25, the more dilute the solution, [p038] the lower being the apparent molecular weight.[60] For sodium chloride, the formula weight, corresponding to the formula NaCl, is 58.5. This would also represent its smallest molecular weight in gas form, consistent with the accepted atomic weights for sodium and chlorine. In ‹aqueous› solution, again, the apparent molecular weight of sodium chloride is found to be less than 58.5, and more than 29.25, the value found depending on the concentration of the solution used. For zinc chloride we have, likewise, in aqueous solution values much less than 136 and tending toward the limit 45, whereas the formula weight for ZnCl_{2} is 136.

These are instances of a very large class of apparent gross discrepancies between the requirements of the Avogadro-van 't Hoff principle and the generally accepted molecular weights of common compounds. There are three ways, in particular, in which one might be inclined to regard such results: in the first place, one might be tempted to consider that van 't Hoff's extension of Avogadro's hypothesis to solutions is justified in a considerable number of cases, but not as a ‹universal› expression, applicable to ‹all› dilute solutions. This seems, indeed, to have been van 't Hoff's own attitude originally. Such a view, since it does not throw new light on the matter, but simply shelves the question of the source of the discrepancy, would be tenable only after all other explanations had been found unsatisfactory.

In the second place, we might be inclined to consider whether a molecule like hydrogen chloride is not dissociated in aqueous solution into two smaller molecules, ‹h›‹cl›, in which hydrogen and chlorine would appear as atoms with the weights ‹h› = 0.5 and ‹cl› = 17.75, which are half as large as the atomic weights determined from a study of volatile compounds of hydrogen and chlorine. If we remember that our atomic weights are confessedly maximum weights, and not minimum weights—although they are almost certainly also the true atomic weights—such a view would be, at least, worthy of some consideration. But, in the first place, it would be extraordinary that we should never have found, in the thousands of [p039] hydrogen derivatives that have been investigated, any compound, the molecule of which, in the gaseous condition, contained a ‹single› such atom of hydrogen, with the weight 0.5, or an ‹uneven› multiple of it: that only ‹even› multiples or pairs ‹h›_{2}, corresponding to the atom H, should always have been found. In the second place, such an explanation of the results of the molecular weight determinations in aqueous solutions given above, would soon lead to difficulties, which make the view altogether untenable. For instance, the molecule of zinc chloride, according to the data given, would have to break down into three molecules and, if these were of uniform composition, we would have to assume chlorine atoms two-thirds or one-third as large as Cl. Since a moment ago we had to assume chlorine atoms one-half as large as Cl, we would have to conclude that the atomic weight of chlorine could be, at most, Cl / 6, which is the largest common divisor of Cl / 2 and Cl / 3. No chemist would seriously consider an atomic weight for chlorine one-sixth as large as the accepted weight, for that would mean that, in all the chlorine compounds investigated in the condition of gases, we have always at least six such atoms occurring together, and otherwise always multiples of six. Consequently such an interpretation of the so-called "abnormal" behavior of solutions of hydrogen chloride, sodium and zinc chlorides, etc., although at one time advanced by some chemists, must be considered as altogether untenable.

A third explanation of the "abnormally" low molecular weights, which certain substances in aqueous solutions possess, is, that the molecules of these compounds are ‹capable of dissociation into smaller molecules of unlike composition›, somewhat like ammonium chloride when it is heated, and that the substances in question are dissociated more or less considerably in this fashion in the solutions under consideration. Hydrogen chloride, for instance, besides existing as such (as HCl), in aqueous solutions, might be capable of dissociating, and actually be dissociated, to a considerable degree into molecules containing either only hydrogen or only chlorine (HCl ⇄ H + Cl); the ‹average› of the weights of the molecules in a mixture of molecules, HCl, H, and Cl, would be ‹less› than 36.5, and, according to the proportion of dissociated and undissociated molecules of hydrogen chloride, the average would lie between the limits 36.5 and (1 + 35.5) / 2, or 18.25. Such an [p040] explanation,[61] made with ‹certain additions and restrictions›, was advanced in 1885 by Arrhenius, a Swedish chemist and physicist, when he learned of the exceptional behavior of these solutions, as noted by van 't Hoff. Although at first this interpretation occasioned considerable criticism, it has maintained itself successfully for twenty years, on the basis of a wide range of accumulated facts, and it has been of remarkable value and benefit in the development of all branches of chemistry and the allied sciences.

«The Theory of Ionization.»—Arrhenius[2] made the simple observation that all those solutions, in which the dissolved compounds seem to have abnormally low molecular weights, are solutions through which an ‹electric current› may be readily passed, they are ‹electrolytes›, whereas the solutions which give normal results (see, for instance, the table on p. 37) do not allow the ready passage of a current, they are ‹nonelectrolytes›.

EXP. The fundamental difference between the two classes of solutions may readily be demonstrated. To water contained in an electrolytic cell, which is connected with a lighting circuit and with an electric lamp, first some alcohol, and later a small quantity of hydrochloric acid are added. The lamp is seen to glow, instantly, when the acid is added.

This simple fact, that the very solutions which give abnormally low molecular weights for the dissolved compounds are also good conductors of electricity, was explained by a theory of ‹electrolytic dissociation› or of ‹ionization›, which Arrhenius had developed[62] from a study of the conductivity of electrolytes. The same fact has aided in establishing this theory which has led to the elucidation of vital problems of ‹electrical conductivity› and to a successful [p041] explanation of the problem of the apparently abnormal ‹osmotic pressures› (and ‹molecular weights›) of electrolyte solutes. It has thus removed the last difficulty in the way of accepting the van 't Hoff-Avogadro Hypothesis (p. 15) as true for all dilute solutions, exactly as the discovery of ‹gaseous dissociation› made it possible to recognize in the original Avogadro Hypothesis a universal truth (p. 36) about gases. And to these results was added, chiefly as the fruit of the work of Ostwald, with the aid of the theory of Arrhenius, the most successful and accurate formulation of the problem of the ‹chemical activity› of electrolytes, known in the history of chemistry.

«Main Assumptions of Arrhenius's Theory of Ionization.»—The main assumptions of the theory of electrolytic dissociation or ionization are the following: (1) When an ionogen is dissolved in water, its molecules are immediately, more or less completely, ‹dissociated by the water› into smaller fragments or molecules of unlike composition. (2) These new molecules are charged with ‹electricity›; the molecules of the one product are charged with ‹positive›, the molecules of the other product[63] with ‹negative electricity›, the unit positive charge being equal in quantity, but opposite in kind, to the unit negative charge; the sum of all the positive charges in a solution is equal to the sum of all the negative charges, and the whole solution is electrically neutral. (3) The dissociation is a reversible reaction, and all electrolytes must be considered to be ‹completely ionized at infinite dilution›. (4) Except for the dependence resulting from the electrical charges and the consequent attractions and repulsions between ions, the ions must be considered independent molecules with their own ‹specific chemical› and ‹physical› properties.

When a current is passed through the solution of an ionogen, the electrified particles carry their charges to the electrodes (see [p042] below). They are called the ‹ions›[64] of the electrolyte; the positively charged ions are distinguished as ‹cations› from the negatively charged ‹anions›, and the electrode toward which the cations move is called the cathode (negative electrode), and the electrode to which the anions move is called the ‹anode› (positive electrode).

The dissociation of hydrogen chloride may be expressed, in the terms of the assumptions made, in the following equation: HCl ⇄ H^{+} + Cl^{−}; that is, hydrogen chloride is dissociated, to a greater or smaller extent and in reversible fashion, into positively charged hydrogen ions H^{+}, and negatively charged chloride ions Cl^{−}, and the charge on each chloride ion is equal in quantity to the positive charge on each hydrogen ion. Zinc chloride is dissociated according to the equation ZnCl_{2} ⇄ Zn^{2+} + 2 Cl^{−}, and, according to (2), the charge on each zinc ion is twice as great in quantity as the charge on each chloride ion, and therefore twice as great also as the charge on each hydrogen ion (see below, p. 58). It is practically certain, according to more recent results, that the ions are combined with water to form hydrates, such as H^{+}(H_{2}O)_{‹x›} and Cl^{−}(H_{2}O)_{‹y›}.[65] This does not modify, essentially, the fundamental assumptions of the theory, but contributes rather to a satisfactory explanation of the rôle of water as an ionizing agent, a question to which we shall return later.

«The Theory of Ionization and the Electron Theory of Electricity and of Matter.»[66]—According to the views held by many leading physicists at the present time, ‹negative electricity› consists of ultimate particles of matter called electrons or corpuscles. The mass of an electron is about 1 / 1000 the mass of an atom of hydrogen, and the electric charge of the electron is equal to that carried by a chloride ion in solution.[67] The atoms of the elements are considered to consist of aggregations of large numbers of electrons in a kind of "shell" or "body" of positive electricity. This positive electricity, in a given atom, is equal in quantity to the total negative charge of the electrons in the atom, the atoms as such [p043] containing no excess of either positive or negative electricity. The number of electrons in the atom of an element is considered to be definite and constant for that element, but the number varies as we go from the atoms of one element to those of a second element, the number increasing with the atomic weight of the element.

One of the most fundamental and most characteristic properties of elements is considered to be the ‹affinity which their atoms show for electrons›; thus, the atoms of metals like sodium and potassium, which are generally called "electropositive" elements,[68] show an enormous tendency ‹to lose one electron› each and to form positively charged particles[69] Na^{-ε} (= Na^{+}) and K^{-ε} (= K^{+}).[70] The atoms of strongly electronegative elements, like chlorine, have a tremendous tendency for ‹gaining› and ‹holding electrons› beyond the number originally in such atoms. Thus, chlorine atoms tend to ‹assume an electron› each; they thereby become ‹negatively charged› particles, Cl^{+ε} ( = Cl^{−}).

On the basis of these views, we have in sodium chloride NaCl a substance, whose molecules contain an atom, Na, with a tremendous ‹tendency to lose an electron›, and an atom, Cl, which has a tremendous ‹affinity for an electron›. It is natural to suppose, then, that ‹both tendencies will be satisfied by the passage of an electron from the sodium to the chlorine atom›, NaCl → Na^{-ε}Cl^{+ε}. Or, if we use the sign + to designate the positive charge produced on an atom by the loss of an electron and the sign − to indicate the charge gained through the assumption of an electron, we have[71]: NaCl is Na^{+}Cl^{−}. Similarly we have in hydrogen chloride H^{-ε}Cl^{+ε} or H^{+}Cl^{−}. It is altogether likely, therefore, that the atoms in a molecule of sodium chloride or of hydrogen chloride already possess electric charges,[72] so that, even while combined, [p044] their tendencies to lose or gain electrons are satisfied. It is also possible that the atoms are held together in the molecule by the electrical attraction of the opposite charges.[73] The force with which opposite electrical charges attract each other depends, as is well known, on the nature of the ‹surrounding medium›. Now, when molecular sodium chloride or hydrogen chloride is dissolved in water (a favorable medium), a decided decrease in the attraction (see p. 62), between the charged atoms within the molecules is brought about, and a process of ‹ionization› results: H^{+}Cl^{−} ⇄ H^{+} + Cl^{−}. The charged particles are called ‹ions› only after they have separated from one another and have become independent molecules, capable, for example, of moving in ‹opposite› directions.

While the atoms of some metallic elements tend to lose a single electron and form ions Me^{+} (‹e.g.› Na^{+}, K^{+}), the atoms of other elements tend to lose two or more electrons, forming bivalent ions, Me^{2+} (‹e.g.› Zn^{2+}, Fe^{2+}, etc.), or trivalent ions, Me^{3+} (‹e.g.› Bi^{3+}, Fe^{3+}), and so forth. Similarly, atoms of the so-called negative elements may assume two or more electrons, forming bivalent ions, X^{2−} (‹e.g.› S^{2−}), and so forth.

«The Validity of the Theory of Ionization.»—In determining the validity of the theory of ionization, we may consider, first, the sufficiency of the explanations which it offers for observed facts and important phenomena. We may then weigh, more critically, by any evidence offering itself, the facts which will enable us to decide between this theory and the older theory of ionization, that of Clausius (p. 51). The latter, although displaced, is still often revived by opponents of the modern theory. Such facts as we will consider are found, first, in the domain of ‹conductivity phenomena›, next, in the ‹osmotic pressure› and related properties of solutions, and, finally, in the study of the ‹chemical activity› of electrolytes (see Chapter V).

«Ionization and Electrical Conductivity.»—Turning our attention first to the field of electrical phenomena, and developing the theory for the present descriptively, we find that the conductivity of a solution depends, according to this theory, on the fact that when two oppositely charged poles are placed in a solution, the [p045] positive charge on the anode attracts all the negative particles within its field of action, and repels all the positive particles, exactly as a positive static charge of electricity would attract a negatively charged pith-ball and repel a positively charged one. In the case of a solution of hydrochloric acid, the negative charge on the cathode would attract the hydrogen ions and repel the chloride ions, and the positive charge of the anode would attract the chloride ions and repel the hydrogen ions. The net result would be a migration of all the chloride ions with their negative charges toward the anode, and of the positively charged hydrogen ions toward the cathode,—a flow or ‹current of electricity› being thus produced. The fact, then, that a current of electricity does readily pass through such a solution of an ionogen, is easily understood on the basis of these views.

EXP. The migration of the ions throughout the whole solution may be demonstrated by the passage of a current through a large U-tube containing a mixture of a cupric salt and a permanganate,[74] placed under some dilute sulphuric acid. The cupric-ion is blue, all ionized solutions of cupric salts, with a colorless negative ion, being blue, while the permanganate-ion is of an intense purple color. In the limb of the U-tube, in which the cathode is placed, a blue zone, containing cupric ions, is soon seen emerging from the purple liquid and rising toward the cathode (see Fig. 8). It will take some time for any cupric ions actually to reach the electrode and be deposited as metallic copper. On the anode side, purple permanganate ions are seen rising toward the positive electrode.

The movement of the electrically charged particles in opposite directions through the solution constitutes an electric current, and such a current has the properties of a current through a [p046] wire—producing, for instance, heat, or being capable of deflecting a magnet placed in its field of action.[75]

«Electrolysis.»—When ions touch an electrode, they are discharged, and with the discharge, are changed chemically, and, according to the electron theory, also materially. Cupric ions, Cu^{2+} or Cu^{−2 ε}, receiving two electrons at the cathode, are precipitated as metallic copper. When a current is passed through a solution of hydrochloric acid, the hydrogen ions, by the pairing of the discharged ions, yield hydrogen gas consisting of molecules, H_{2}; the chloride ions are converted at the anode into chlorine Cl_{2}. The formation of these molecules, X_{2}, may be variously interpreted, as resulting either from the union of two discharged or neutral atoms, or as consisting, for instance in the case of hydrogen, in the discharge of a hydrogen ion H^{+}, at the negative electrode, ‹followed by the assumption of a negative charge or electron by the neutral atom›, the new ion H^{−} combining at once with the positive ion H^{+} to form the molecule H^{+}H^{−}. At the positive pole, by an analogous recharging of a discharged negative chloride-ion Cl^{−}, we should obtain a positive ion,[76] Cl^{+}, and immediately the formation of Cl_{2} or Cl^{+}Cl^{−} would follow. Helmholtz advocated the latter conception of the action at the electrodes, and, more recently, J. J. Thomson[77] brings forward, as a very important argument in favor of it, the fact that iodine, which according to vapor density determinations dissociates into monatomic molecules at high temperatures (I_{2} ⇄ 2 I), becomes, simultaneously with this dissociation, also an excellent ‹gaseous conductor of electricity›, as would be anticipated from a dissociation, I^{+}I^{−} ⇄ I^{+} + I^{−}. This fact is emphasized here, because in it we seem to have a case of ‹conductivity› coinciding with ‹gaseous dissociation›, the existence of which is recognized by the universally accepted laws of gases, and differing in no important respect from the dissociation assumed for conductors in solution.

«Conductivity and Dilution.»—The passage of a current through an electrolyte consists, then, in a transfer of electricity by material particles, the ions. According to the theory of ionization only [p047] the ionized portion of an electrolyte can carry the current at any moment, and, consequently, a ‹given weight› of an ionogen should, under comparable conditions, be the more efficient as a conductor, the more completely it is dissociated into ions.

If the conductivity of a given weight of hydrogen chloride, for instance, is measured under comparable conditions, it should be found to be greater, the more completely the acid is ionized. Now, in aqueous solutions, hydrogen chloride ionizes under the influence of the ‹solvent water› (pp. 41, 61), and the theory would lead us to anticipate that the greater the proportion of water used, the more extensively will it ionize the acid. Consequently, the addition of water to a given weight of acid should increase the latter's efficiency as a conductor. This conclusion has been fully verified by exact methods of measurement and may be readily demonstrated by the following series of experiments:

EXP.[78] An electrolytic cell, having the shape of a parallelopipedon and a capacity of about one liter, is fitted with electrodes of copper, which reach from the bottom to the top of the cell and are connected with a storage cell and an ammeter. The cell is first filled with distilled water: no perceptible current passes through the water and the latter is therefore practically a nonconductor. The cell is then emptied by means of a siphon and 20 c.c. of 4-molar hydrochloric acid is brought into it. The ammeter shows that a definite current passes through the solution (0.17 ampere in an experiment[79] with a cell 4.6 cm. wide and 11.5 cm. long, with copper electrodes 4.6 cm. broad and 21 cm. high). (See Fig. 9, p. 48.) [p048]

The conductivity of a solution, like that of a metal conductor, is the reciprocal of its resistance. Since, according to Ohm's law,[80] the current for a ‹given potential› is inversely proportional to the resistance, the current is also directly proportional to the conductivity. The resistances of the metal connections and of the ammeter in the experiment are very small compared with the resistance of the solution, and they may be considered negligible for our purpose. Thus, ‹the current indicated by the ammeter is a closely approximate measure of the conductivity of the solution›. Now, if a volume of water (20 c.c.) equal to the volume of acid, were to be added to the latter, the cross section through which the current flows from plate to plate would be ‹doubled›, and, since the conductivity of a liquid conductor, like that of a metal, increases proportionally to the cross section, the current should be doubled by the change in this one factor. On the other hand, the concentration of the conducting acid is now ‹one-half› of the original concentration, and this should in turn reduce the conductivity of the solution to one-half. Consequently, if there were no further change in the electrolyte, the original conductivity should be maintained when the acid is thus diluted. But, according to the theory of ionization, as has just been shown, the addition of [p049] water to a given weight of hydrochloric acid ‹should increase the proportion of ionized acid›, and since the ions are the carriers of the current, the ‹conductivity› of the solution should be ‹increased› because of this change in the ‹composition› of the electrolyte. Experiment shows that such is the case.

EXP. 20 c.c. of water is added to the 20 c.c. of 4-molar acid in the cell, and the mixture is stirred. The current is decidedly increased (from 0.17 to 0.22 ampere in the experiment under discussion). If 40, 80, 160 and 320 c.c. of water are added in succession to the contents of the cell, the conductivity is ‹increased by every addition of water›. But, while each addition dilutes the acid to one-half the previous concentration, the ‹increase grows proportionally smaller and smaller with increasing dilution›. In the following table, "Ratios I" are the ratios of the observed conductivities ‹to the original conductivity›, "Ratios II" the ratio of each observed conductivity to the ‹preceding one›.

‹Concentration› ‹Observed› Ratios I. Ratios II.
‹of Acid.› ‹Conductivity.›[A]

4-molar 0.17 1 1.
2-molar 0.22 1.30 1.30
1-molar 0.26 1.53 1.18
0.5-molar 0.30 1.76 1.15
0.25-molar 0.31 1.83 1.04
0.125-molar 0.32 1.88[B] 1.03

TABLE NOTES:

A. This is an artificial scale (see text) of conductivities,
and does not represent reciprocal ohms, the standard units of
conductivity.

B. In the exact data on the conductivities of 4-molar and
1/8-molar HCl (Kohlrausch and Holborn, ‹Leitvermögen der
Elektrolyte› (1898) p. 160), the ratio 348 / 181.5, or 1.92,
is found, in place of 1.88 as observed.

We should expect, further, that the increase in conductivity, being dependent on the increased dissociation of a finite quantity of electrolyte, should tend towards a ‹limit›, a maximum conductivity being reached when (practically) all the acid is ionized. As a matter of experience, the conductivity of a given quantity of an acid or other ionogen does tend toward a ‹limit›. In the experiment just made, the conductivity of the acid increases very rapidly at first, as the 4-molar acid is diluted by water; but the increase in conductivity with the succeeding dilutions grows ‹smaller› and ‹smaller› and the conductivity is plainly approaching [p050] a limit (see the ratios I and II in the table). For hydrochloric acid at 18°, the limit for one mole[81] (36.5 grams HCl) at infinite dilution, as deduced from the curve of conductivities at finite dilutions, is 384 reciprocal ohms.[82]

«Degree of Ionization of an Electrolyte.»—The conductivity of a given weight of an electrolyte, for instance of its gram-equivalent weight, depends, then, at a given temperature on the extent to which it is ionized, the ions being the only carriers of the current in a solution of an electrolyte. The conductivity will also depend on the friction which the ions must overcome in moving through a solution, but, for sufficiently dilute solutions in a given solvent, the friction may be assumed to be approximately constant for given ions. For such solutions, then, the conductivity of a given weight of a given electrolyte at a given temperature may be said to depend wholly on the extent to which the electrolyte is ionized. Thus, the proportion of ionized electrolyte in a solution may be determined by measuring the conductivity. ‹The extreme limit of its conductivity, calculated for infinite dilution, represents complete ionization› of the electrolyte according to a fundamental postulate (§ 3, p. 41) of the theory of Arrhenius, and the ratio of the conductivity in a given solution to the conductivity of the same weight of electrolyte at infinite dilution represents then the ‹proportion of ionized electrolyte to the total electrolyte› used. This proportion is called its ‹degree of ionization› (commonly designated by α). If we call Λ_{‹v›} the conductivity of a gram-equivalent weight of an electrolyte in a given solution, and Λ_{∞} the limit of its conductivity for infinite dilution, then the degree of ionization is found from α = Λ_{‹v›} / Λ_{∞}. [p051]

The method of calculation of α in a specific case may be illustrated as follows: the resistance of a cube of 1 cm. edge of a solution of hydrochloric acid, which contains 1.825 grams hydrogen chloride in a liter, is found to be 55.55 ohms at 18°. Its conductivity then is 1 / 55.55 reciprocal ohms. Now, 1.825 grams of hydrogen chloride is 1.825 / 36.5 or 1 / 20 gram-equivalent of the acid; a whole gram-equivalent of the acid would be contained in 20 liters or 20,000 c.c. Then Λ_{‹v›} = (1 / 55.55) × 20,000, or 360 reciprocal ohms. If we use the value at infinite dilution given above, α = 360 / 384, or 93.75%. That is, 93.75% of the hydrochloric acid is present in the ionized condition in such a solution, and 6.25% is not ionized.

By making the assumption that ‹at infinite dilution electrolytes are completely ionized›, and by taking the ratio which the equivalent conductivity of a given solution of an electrolyte bears to the maximum limit-value (calculated for the conductivity at infinite dilution) ‹to be the degree of ionization of the electrolyte›, as just explained, the theory of Arrhenius has thus made it appear possible ‹to determine experimentally the proportion of ionized electrolyte present›.

It is a significant fact that the equivalent conductivity of hydrochloric acid is ‹close to its limit even at finite dilutions›, and that the same relation holds for the strong acids and the strong bases, in general, and for most salts. But the equivalent conductivity of weak acids, like acetic acid, and of weak bases, like ammonium hydroxide, in finite dilutions is still far removed from the limits which may be calculated for infinite dilutions. Arrhenius was led then to the further important conclusion that, in the case of the first electrolytes mentioned, a ‹very large proportion of the electrolyte must exist in the ionized form at finite concentrations›, their equivalent conductivities having almost reached the limit characteristic of infinite dilution.

«Clausius's Theory of Ionization and the Modern Theory.»—It is in these conclusions—in particular that the proportion of ionized electrolyte ‹may be determined experimentally›, and that frequently a ‹large proportion is found to be ionized at finite concentrations›,—that the ‹modern theory of ionization differs from the older theory of Clausius›. The former is an elaboration of the latter, and some opponents[83] of the modern theory still uphold the latter as offering an adequate explanation of the phenomena of conductivity. All facts, then, in particular, which confirm the validity of the ‹conception of the degree of ionization›, as introduced by Arrhenius, [p052] must be considered as criteria favoring his theory, specifically. The development of chemistry in the last twenty years is replete with such evidence and we shall meet it in many connections throughout our work.

Clausius[84] also assumed dissociated molecules or ions to be the real carriers of electricity in the passage of a current through the solution of an electrolyte, but he assumed only a ‹minute quantity of these molecular fragments or ions to be free at any moment›, their existence being supposed to be transitory and dependent in particular on exchanges of atoms between molecules. As a result of the oscillations of the atoms composing a molecule, oscillations comparable with the motions of molecules assumed in the kinetic theory of gases, molecules were considered by Clausius occasionally to reach such a condition of instability, that they dissociated into smaller particles; since the atoms were supposed to be held in a molecule by attractions of electrical charges on the atoms (theory of Berzelius), the fragments of the molecule would carry the charges, positive and negative respectively, which they possessed in the molecule. Such a breaking up or dissociation of molecules was, further, supposed to occur with particular ease during the collisions of molecules, the electrical attractions and repulsions of the charged atoms favoring, at such moments, an ‹exchange of atoms›. During the exchange, the atoms were considered to be free molecules, charged with electricity—essentially ions,—capable of moving under the influence of electrical forces and of thus carrying a current. Finally, such ions were supposed, in part, to escape recombination, and to remain free, until each ion either collided and combined with an ion of opposite charge, or collided with a molecule and displaced an atom of the same charge from that molecule, a new ion being thus liberated. The theory, as usually interpreted, assumed the existence of only a very small quantity of such free ions, that being all that was supposed to be required to explain the facts known at the time it was advanced.

In what follows, we shall confine the discussion strictly to such contrasts between the two theories as grow out of a consideration of the phenomena of conductivity, and particularly consider some evidence which is directly concerned with the conductivity of solutions.

In the first place, if the formation of ions occurs primarily during the exchange of atoms in ‹collisions of molecules›, then, as Whetham[85] has shown, the specific conductivity (of 1 cm.^3) of an electrolyte, like hydrochloric acid, must increase with the concentration and must increase, approximately, as a function of the ‹third› power of the concentration. The more concentrated the solution, the more frequent the collisions between the dissolved molecules must be. As a matter of fact, as shown in the following table, the conductivity [p053] increases a ‹little less› than proportionally to the ‹first› power of the concentration—‹which is in conflict with the assumption made in the hypothesis of Clausius›, but in perfect agreement with the hypothesis of Arrhenius. The small ‹decrease› with increasing concentration, in the simple ratio between conductivity and concentration, is due to the decreasing degree of ionization in the more concentrated solutions, as demanded by the hypothesis of Arrhenius.

The table gives, in the first column, the specific conductivities of hydrochloric acid at 18°, and, in the second column, the concentrations; these concentrations are expressed in moles or gram-equivalents per cubic centimeter; the last column gives the ratio of conductivity to concentration.

Conductivity Conductivity
of 1 c.c. Concentration ─────────────
Concentration.

0.00370 0.00001 370
0.00734 0.00002 367
0.01092 0.00005 364
0.01800 0.00005 360
0.03510 0.00010 351

Furthermore, facts admitted by Clausius to be inexplicable by his own assumptions receive, in the theory of Arrhenius, at least a quantitative formulation borne out by a mass of corroborative evidence. The difference in conductivity between pure water and sulphuric acid is such a fact, mentioned by Clausius. Determinations of the ionization of sulphuric acid and of water, by the conductivity methods which are based on the theory of Arrhenius, show that, while sulphuric acid is very considerably ionized (see p. 104), water is scarcely ionized at all. The ionization of water (see p. 104) has been determined quantitatively by at least four independent methods of examination,[86] and, minimal as the ionization is, the results agree so well with each other that van 't Hoff[87] was led to write: "If one is not previously convinced of the correctness of the theory of electrolytic dissociation, hardly any result won by means of it is so convincing, as the agreement between the conclusions reached in completely different ways as to the degree of dissociation of water itself. After such an agreement, it is hardly conceivable that the basis on which all these results rest should further be altered."

«Mobilities or Partial Conductivities of Ions: Principle of Kohlrausch.»—If ions have a separate existence, each kind of ion would be expected to move through a solution under a given electrical force at a given temperature with its own specific speed, the speed being presumably dependent on the nature of the ion as well as on the weight of water combined with it and dragged with it through the solution (p. 42). [p054]

Such relative speeds of ions may be demonstrated by means of an experiment: the motion of the hydrogen ions, formed by the ionization of hydrochloric and other acids, may be observed by their action on a reddened (alkaline) solution of phenolphthaleïn, which is decolorized by them; and the motion of the hydroxide ions, formed by the ionization of sodium hydroxide and other bases, may be followed by their action on colorless phenolphthaleïn, which turns red in their presence. The hydrogen and the hydroxide ions are the fastest, in aqueous solutions, and their speeds are compared in the next experiment with that of blue cupric ions, which have a speed roughly the same as that of many common ions. In this experiment the hydrogen ions are readily seen to move about twice as fast as the hydroxide ions and five to six times as fast as the cupric ions.

EXP.[88] Five grams of agar-agar are dissolved in 250 c.c. of boiling water. To 100 c.c. of the hot solution, 32 c.c. of a saturated solution of potassium chloride and about 1 c.c. of phenolphthaleïn solution are added, together with enough of a solution of potassium hydroxide, added drop by drop, to produce a deep red tint in the phenolphthaleïn. Of this mixture 50 c.c. is treated with dilute hydrochloric acid, added drop by drop, until the red color is just discharged, and then an excess of acid, equal in amount to the quantity used to neutralize the 50 c.c., is added to the mixture. This colorless solution and 50 c.c. of the red solution are poured, while still warm, into the two parts of a wide U-tube, slowly and at equal rates, so that the level on the two sides remains the same. In this way it is possible without difficulty to have the solution on one side red (alkaline) and on the other side colorless (acid). The agar-agar is allowed to congeal, and then a mixture of 0.5 c.c. of hydrochloric acid, (sp. g. 1.12), 6 c.c. of saturated cupric chloride solution and 20 c.c. of water is poured over the red half, and a mixture of 20 c.c. of saturated [p055] potassium chloride solution and 2 c.c. of 10% potassium hydroxide solution is poured over the colorless half. The U-tube is surrounded by ice water during the passage of the current, and the cathode is placed in the solution on the colorless side. In Fig. 10 the U-tube is shown when first charged (on the left), and after the current has been running for a short time (on the right).

The conductivity of a solution must be made up, therefore, of the sum of the shares which the positive ions and the negative ions, respectively, take in carrying the current. This principle was first advanced by Kohlrausch. The share of each kind of ion in conducting a current may be determined, for hydrochloric acid for instance, in the following way: A porous diaphragm may be used to divide the solution in an electrolytic cell into two halves, the concentration of the acid being the same in both halves (represented, as indicated in Fig. 11, by 15 molecules[89] of ionized acid in each half). A measured current is passed through the solution, say, sufficient to liberate 3 molecules of hydrogen H_{2}, and 3 of chlorine Cl_{2}, corresponding to 6 ‹ions› of each, and the concentration of the acid in each half is then again determined by analysis. Say it is found to correspond to 14 molecules of hydrochloric acid in the half of the solution on the side of the cathode and 10 molecules in the half on the side of the anode (see Fig. 12). Then the anode half has lost 5 ions of hydrogen, which must have passed through the diaphragm toward the cathode and taken the place of five of the six hydrogen ions discharged at the cathode. Similarly, the solution around the cathode has lost one chloride ion, which must have passed through the diaphragm toward the anode, and the hydrogen-ion corresponding to it, remaining on the right side without a compensating negative ion, must be the sixth hydrogen-ion discharged at the cathode. In other words, five hydrogen ions passed to the right, while one chloride ion passed to the left. The hydrogen ions then carried five-sixths of the current through the diaphragm, and consequently through the solution, and the chloride ions only one-sixth of the current. Since the solutions were of equal concentration to start with, the hydrogen ions have moved ‹five times as fast› toward the cathode as the chloride ions have moved toward the anode.

The equivalent conductivity of 0.1-molar hydrochloric acid is 351 at 18°, and experiment shows that the hydrogen-ion carries 84% of the current, the chloride-ion only 16%. The conductivity may then be considered to be the [p056] sum of the share the hydrogen-ion has in carrying the current, ‹i.e.› 0.84 × 351, or 295, and of the share of the chloride-ion, 0.16 × 352.5, or 56. These values may be called the ‹equivalent partial conductivities› or ‹mobilities› of the ions in this solution.

In a similar way, the conductivity of every solution of an electrolyte may be shown to represent the sum of the mobilities of the ions carrying the current (‹principle of Kohlrausch›). The limit of the conductivity of one equivalent of an electrolyte is the sum of the mobilities of the ions composing the electrolyte. The frictional forces being constant for infinitely dilute solutions, at a given temperature, an ion will always show the same mobility, irrespective of the nature of the ion of opposite charge, with which it forms the electrolyte. We may then put Λ_{∞} = (‹l›^{+}_{∞} + ‹l›^{−}_{∞}), if ‹l›^{+}_{∞} and ‹l›^{−}_{∞} are used to designate the limits of the mobilities of gram-equivalents of the positive and negative ions forming the electrolyte. The following table[90] gives the limits of the mobilities for gram equivalents of some of the most important ions at 18°.

‹Limits of Mobilities of Common Ions at› 18°.

K: 65.3 ½ Ca: 53.0 I: 66.7
Na: 44.4 H: 318.0 NO_{3}: 60.8
(NH_{4}): 64.2 OH: 174.0 C_{2}H_{3}O_{2}: 33.7
Ag: 55.7 Cl: 65.9 ½ SO_{4}: 69.7

For quite dilute solutions, in which the friction may be assumed to be approximately constant, the conductivity will depend, not only on the mobilities of the ions, which may be taken to be the same as for solutions of extreme dilution, but also ‹on the proportion of electrolyte that is ionized›, ‹i.e.› on the degree of ionization, α. Then Λ_{‹v›} = α (‹l›^{+}_{∞} + ‹l›^{−}_{∞}), which is an elaboration of the original equation given on page 50.

Now, Kohlrausch discovered the principle of the summation of the mobilities of ions a number of years before the theory of Arrhenius was advanced, and the proportion in which the ion is present in a given solution being unknown, the effect of what is here known as the degree of ionization was included empirically in the value of the mobility. It is not surprising, then, that an ion was found to have approximately the same mobility ‹only› in solutions of the same concentration ‹of strictly analogous and closely related salts›, which, according to present methods of investigation, are ‹now› found to have approximately the same degree of ionization. For instance, the mobility of the gram-equivalent of the chloride-ion was found to be approximately the same, 47.3 and 50.5 respectively, in molar solutions of sodium and potassium chloride at 18°, no account being taken of the degrees of ionization. However, the degrees of ionization of the two salts are approximately the same, 66.9% and 74.9% respectively, and might be ignored in a comparison of the conductivities, without affecting the result of the comparison in any marked way. [p057]

When the conductivities of unlike electrolytes are compared, the ‹introduction of the conception of the degree of ionization› (by Arrhenius,) into Kohlrausch's principle of the independent conductivities of specific ions, shows most striking results and ‹demonstrates the value of the new conception›. For instance, the equivalent conductivity of potassium chloride at 18° in 0.075 molar solution is 113.8 reciprocal ohms and the partial conductivity of the chloride-ion in the solution is 57.4. But the conductivity of an equivalent solution of ‹mercuric chloride› at 18° is only 1.51, which is very much less than the conductivity of the chloride-ion alone in the potassium chloride solution. Now, mercuric chloride, according to investigations of its conductivities and of its effect in depressing the freezing-point of water,[91] is one of a very few salts that are difficultly ionizable (p. 107); according to the data mentioned, it is ionized, at most, to the extent of 2.5 per cent in the solution in question, whereas 87.5 per cent of the potassium chloride is ionized in such a solution. When the difference in the degree of ionization is taken into account, the conductivity which mercuric chloride ‹should show› may be calculated, ‹on the assumption that the chloride-ion has the same mobility› in the two solutions, but that there is less of it in the mercuric solutions. We put Λ_{HgCl_{2}} = α (‹l›_{Hg} + ‹l›_{Cl}) = 0.025 (48 + 65.9) = 2.8. We thus find that the conductivity of the mercuric chloride should be, approximately, only 2.8 reciprocal ohms, which is of the same order as that found (1.51).[92]

In the same way, when we compare the conductivity of a strong acid, like hydrochloric acid, with that of a weak acid, like acetic acid—the conductivity of 0.1 molar hydrochloric acid is 351, of 0.1 molar acetic acid only 4.6—the principle of the specific, characteristic mobility of the hydrogen-ion, which is present in both solutions, has significance only if we take into account the very different concentrations of the hydrogen-ion in the two solutions, ‹resulting from the different degrees of ionization of the two acids›—91% for the hydrochloric and only 1.7% for the acetic acid. The same relations hold in the comparison of the conductivity of a solution of a strong base like sodium hydroxide with that of an equivalent solution of a weak, i.e. much less ionized base like ammonium hydroxide, or in comparing the conductivity of a ‹weak acid› or a ‹weak base› with the conductivities of their ‹much more highly ionized salts›.

‹In all these cases the use of the conception of the degree of ionization of the electrolytes› makes possible a much broader and more general application of the principle of the independent migration or mobility of the ions than was possible before the theory of Arrhenius was proposed, and marks a distinct advance in the theory of conductivity, over what was possible on the basis of the theory of Clausius. [p058]